Match the verbal description with the part of the graph it describes.

Use the Divergence Theorem to evaluate S F · dS, where F(x, y, z) = z2xi + y3 3 + sin z j + (x2z + y2)k and S is the top half of

GenaCL600 [577]

Close off the hemisphere $S$ by attaching to it the disk $D$ of radius 3 centered at the origin in the plane $z=0$ . By the divergence theorem, we have

$\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV$

where $R$ is the interior of the joined surfaces $S\cup D$ .

Compute the divergence of $\vec F$ :

$\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2$

Compute the integral of the divergence over $R$ . Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

$\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}$

So the volume integral is

$\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5$

From this we need to subtract the contribution of

$\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S$

that is, the integral of $\vec F$ over the disk, oriented downward. Since $z=0$ in $D$ , we have

$\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k$

Parameterize $D$ by

$\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath$

where $0\le u\le 3$ and $0\le v\le2\pi$ . Take the normal vector to be

$\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k$

Then taking the dot product of $\vec F$ with the normal vector gives

$\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v$

So the contribution of integrating $\vec F$ over $D$ is

$\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4$

and the value of the integral we want is

(integral of divergence of F) - (integral over D) = integral over S

==> 486π/5 - (-81π/4) = 2349π/20

5 0

3 years ago

PLEASEEEE HELP ME OUT!

lubasha [3.4K]

There are no real solutions to that equation.

8 0

2 years ago

What is 10% of 50? A. 6.75 B. 7.5 C. 10.5 D. 5.25

nirvana33 [79]

Its 5. Its none of those.

10% of 50.00 = 5.0000

5 0

3 years ago

Which of the following is most likely the next step in the series ?

GalinKa [24]

$\huge \boxed{\mathfrak{Question} \downarrow}$

Which of the following is most likely the next step in the series ?

$\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}$

The next series is most probably going to be option D. 
We can see from the series that, figure 1 has 4 points marked on the circle from which they formed a rectangle.
In the next figure, there are 3 points marked which together when joined forms a triangle.
Then the next circle has 2 points marked on it which makes a straight line.
So, obviously the next figure/circle in the series should have only 1 point marked on it. This won't form anything as you can't join just 1 point with something.

7 0

3 years ago

Read 2 more answers

Please help!! 15 points for the answer.

Anarel [89]

A) (5s + 2 / 4) x (4) = Perimeter
B) ( s + 9) x (2) + ( 3s - 5 ) x (2) = Perimeter
C) 22 + 48 = 70

4 0

3 years ago